Question

An air chamber of volume $V$ has a neck of cross sectional area '$a$' into which a light ball of mass ' $m$ ' just fits and can move up and down without friction. The diameter of the ball is equal to that of the neck of the chamber. The ball is pressed down a little and released. If the bulk modulus of air is $B$, the time-period of the oscillation of the ball is $2 \pi m ^{ x } V ^{ y } B ^{ z } a ^{ q }$. The value of $\frac{ xy }{ q }$ is_______.

Answer 1

Change in volume, $\Delta V = ax$
excess pressure, $\Delta P =- B \frac{\Delta V }{ V }$
Restoring force $=$ excess pressure
$\times$ cross-sectional area
$\Rightarrow F =- Ba \frac{\Delta V }{ V }$
$=\frac{- Ba ^{2}}{ V } x [\because \Delta V = ax ]$
In SHM, F = $- kx$
$\therefore k =\frac{ Ba ^{2}}{ V }$
$ T =2 \pi \sqrt{\frac{ m }{ k }}=2 \pi \sqrt{\frac{ mV }{ Ba ^{2}}} $
$\Rightarrow x =\frac{1}{2}, y =\frac{1}{2}, z =\frac{-1}{2}, q =-1 $
$\Rightarrow \frac{ xy }{ q }=\frac{\left(\frac{1}{2}\right)\left(\frac{1}{2}\right)}{(-1)}=-0.25$